Put the onion in a food processor and pulse until it resembles rice. Heat 1 tbsp oil in a large frying pan, tip in the onion and fry for 10 mins until soft and just golden. Stir in the garlic and fry for 1 min more. Tip the mixture into a large bowl.
Put the mushrooms in the food processor and pulse until they also resemble rice. Heat the remaining oil in the pan, add the mushrooms and fry for 10-15 mins, until any liquid that’s released has evaporated and the mushrooms are tender. Tip into the bowl with the onions.
Put the tofu, beetroot, beans, tomato purée, mayonnaise, miso, flour and flaxseed in the food processor with some seasoning and blitz until smooth. Scrape into the bowl with the veg. Add the pecans and half the rice to the food processor and pulse until finely chopped, or a little chunkier, if you prefer. Tip into the bowl with the veg and tofu mixture, along with the rest of the rice.
Use your hands to shape the mixture into patties roughly the same size as your burger buns. The patties will be delicate, so chill for at least 30 mins before cooking. If you want to barbecue them, freeze for 30-45 mins first until firm.
Heat a frying pan over a medium heat or light the barbecue, and brush both sides of the patties with a little oil. Cook for 5-8 mins on each side, turning once after a crust has formed, until charred and hot through to the centre. Avoid moving or turning too often, as they will be more delicate than meat patties. Top with the cheese, if using, for the final minute or so (see tip, below). Toast the buns on the barbecue or in the pan, then spread each with a little chilli jam. Fill with a few lettuce leaves, the patties and some smashed avocado mixed with a squeeze of lime.